Kinematic Geometry of Surface Machinin Episode Episode 9 doc

256 Kinematic Geometry of Surface Machiningprecisely specify geometric parameters of the active part of a cutting tool, all the elementary motions that compose the resultant motion of th

The Geometry of the Active Part of a Cutting Tool 253

active part of a cutting tool in many cases is measured in reference planes

con-figuration of which depends upon tangent planes to the surfaces R s and C s The

effect of the cutting edge torsion onto the material removal process in metal

cutting has not yet been profoundly investigated

A cutting edge can be considered as a line of intersection of three surfaces: the

generating surface T of the cutting tool, the rake surface R s, and the clearance

surface C s The equation of the cutting edge can be derived as a result of mutual

consideration of the equation of one of three pairs of surfaces: (a) rrs(U V rs, rs) and

rcs(U V cs, cs), (b) rT(U V T, T) and rrs(U V rs, rs), or (c) rT(U V T, T) and rcs(U V cs, cs)

The solution to any of three pairs of equations can be reduced to the equation

of the cutting edge, which yields representation in matrix form:

rce rce ce

T ce

T ce T t

( )( )( )

where t ce denotes the parameter of the cutting edge

In a particular case, length S ce of the cutting edge can be chosen as the

parameter of the cutting edge (that is, t ce≡S ce) Under such a scenario,

tor-sion τce of the cutting edge can be computed from

τce ρce ce

ce

ce ce

ce ce

2 2

3 3

where the sign of the torsion τce is not in compliance with the direction of

the angle of inclination λs

6.2.6 Diagrams of Variation of the Geometry of the Active

Part of a Cutting Tool

Analytical methods for the computation of actual values of the geometry of the

active part of a cutting tool are accurate They are capable of computing the

dis-tribution of a geometrical parameter of a cutting tool both at a given point of the

cutting edge in different reference cross-sections or within the active part of the

cutting edge in similar cross-sections Results of such computations are accurate

and are of critical importance to a tool designer For the preliminary analysis of

the geometry of the active part of a cutting tool, the implementation of diagrams

of variations of the geometrical parameters have proven useful

Distribution of the function tanγ of the rake angle in different

refer-ence planes through the point M within the cutting edge of a form-cutting

tool is shown in Figure 6.14 Once the rake angle in two different reference

planes is determined, then the distribution of the function tanγ follows the

circle The circle constructed on any two known vectors through the point M

enables easy determination of the function tanγi in any direction through

The Geometry of the Active Part of a Cutting Tool 255

For analysis of the distribution of the function cot g, Figure6.16 is helpful

[4] Two known values of the function cot g in known corresponding

direc-tions yield construction of the straight line AB Ultimately, the actual value

of the function cot giin the reference plane through a current direction is

specified by the corresponding point within the straight line AB.

All the diagrams are in perfect correlation with the results of the

analyti-cal computations

6.3 Geometry of the Active Part of Cutting Tools

in the Tool-in-Use System

When machining a part surface, the actual direction of the primary motion,

as well as of the feed-rate motion, can differ from the assumed directions of

these motions, say in the tool-in-hand system Moreover, the actual

kinemat-ics of a machining operation can be made up not only of the primary and the

feed-rate motions, but also of motions of another nature (for example,

vibra-tions, orientation motions of the cutting tool [see Chapter 2], etc.) In order to

cot γ o

cot γ C

cot γ o cot γ E cot γ F

A D C

256 Kinematic Geometry of Surface Machining

precisely specify geometric parameters of the active part of a cutting tool, all

the elementary motions that compose the resultant motion of the cutting tool

relative to the work must be taken into consideration

There are two possible ways to represent the machined surface P First, the

machined surface P can be considered as an enveloping surface to

consecu-tive positions of the generating surface T of the cutting tool when the cutting

tool is moving relative to the blank Second, the machined surface P can be

considered as a set of discrete surfaces of cut P se

At an instance of time when the surface P is generated, both the surface T

and the surface P se are tangent to P either at point K or along the

characteris-tic curve Because of this, the tool-in-hand reference system can be associated

either with the assumed surface of the cut or with the cutting tool The two

options are identical in the sense of the tool-in-hand reference system

When machining a part surface, it is necessary to consider the kinematic

geometric parameters of the active part of the cutting tool in a reference

sys-tem associated with the surface of the cut For this purpose, the tool-in-use

reference system is used

Commonly, rake surface R s as well as clearance surface C s of a cutting

tool are shaped in the form of three-dimensional surfaces having complex

geometry Due to this, consideration of the surfaces R s and C s at a distinct

point of the cutting edge is required Contact of the cutting wedge with the

work is considered at a distinct point of the cutting edge Because size of the

area of contact of the cutting edge and the work is small, the rake surface

as well as the clearance surface are locally approximated by corresponding

planes, by the planes that are tangent to the surfaces R s and C s at the point

of interest of the cutting edge

Generally speaking, the geometry of the active part of a cutting tool must

be determined for an elementary cutting edge of length dl (that is, in

dif-ferential vicinity of the point M within the cutting edge) It is also necessary

to consider the geometry of the active part at a given instant of time, say for

the vector VΣ of known magnitude and direction Such an approach would

enable one to determine the distribution curves of geometric parameters

within the cutting edge and the distribution curves of geometric parameters

in time In order to perform such an analysis, a generalized method of

com-putation of geometry of the active part of a cutting tool is necessary

In particular cases, actual values of geometric parameters of the active

part of a cutting tool can impose certain constraints onto parameters of

kinematics of the machining operation For example, variation in the actual

value of geometric parameters either within the cutting edge or in time

may impose restrictions on the parameters of feed-rate motion, of

orienta-tion moorienta-tion of the cutting tool, and so forth If parameters of kinematics of

the machining operation exceed the limits, then the machining operation

is not feasible

The capability to determine critical feasible values of parameters of

geom-etry of the active part of a cutting tool is critically important for the tool

designer

The Geometry of the Active Part of a Cutting Tool 257

6.3.1 The resultant Speed of relative Motion in the Cutting

of Materials

As follows from the above analysis, the direction of the resultant speed VΣ

of relative motion in the cutting of materials is a critical issue for establishing

the tool-in-use reference system Usually, relative motion of the cutting tool

is of a complex nature In the general case of surface machining, this motion

is composed of the actual primary motion Vp, the surface generation motion

Vgen, one or more feed-rate motions Vf i., the orientation motions of the first

Vor I and of the second Vor II kinds, and of other motions This yields the

fol-lowing equation for vector VΣ:

m

.

(6.65)

where Vp is the vector of the primary motion, Vgen is the vector of the motion of

surface generation, Vf i. is the vector of the feed-rate motion, n is the total

num-ber of feed-rate motions, Vor I is the vector of the orientation motion of the first

kind of the cutting tool, Vor II is the vector of the orientation motion of the second

kind of cutting tool, Vj is the j elementary relative motion of the cutting tool,

and m is the total number of elementary relative motions of the cutting tool.

When determining the vector VΣ, vectors of all particular relative motions,

those that significantly affect the VΣ must be taken into account Relative

motions, those that cause sliding of the surface P or the generating surface T

of the cutting tool over itself must be incorporated as well

Motions Vor I and Vor II of orientation of the cutting tool, as well as the

feed-rate motions Vf i., are usually significantly smaller compared to the primary

motion Vp However, all must be incorporated for determination of the vector

VΣ In particular cases, some of these motions are comparable with the motion

VΣ Moreover, in special cases, they can even exceed the primary motion Vp

When cutting a material, vibration of the cutting tool is often observed The

vibration may result in positive and negative clearance angle (Figure 6.17a) For

certain frequencies and magnitudes of the vibration, neglecting the vector of

vibration Vvib is not allowed [1,3,13] Due to vibrations, the rake and the

clear-ance angles vary within a certain interval ±σo The current value of the angle

sponding rake angle γo raises to the range of γo′ =γo+δo At this instant, the

clearance angle αo reduces to αo′ =αo-δo If the vector Vvib is directed

oppo-sitely, then the corresponding rake angle γo and the clearance angle αo can be

computed from the equations γo′′=γo-δo and αo′′=αo+δo (see Figure6.17b)

When a partly worn cutting tool is used, then the clearance angle within a

narrow land on the clearance surface next to the cutting edge reduces to 0°

The Geometry of the Active Part of a Cutting Tool 259

The surface of cut P se can be represented as a locus of consecutive

posi-tions of the cutting edge that travels with the resultant speed VΣ relative to

the work The plane of cut is tangent to the surface of cut P se at the point of

interest within the cutting edge In a particular case, the surface of the cut

and the plane of the cut are congruent The last case is the degenerated one

The main reference plane P re is perpendicular to the vector VΣ The

work-ing plane P fe is the plane through the directions of the primary motion, and

of the feed-rate motion Due to this, the working plane P fe is perpendicular

to the main reference plane P re The tool back plane P pe is perpendicular to

the reference planes P re and P fe

Other reference planes are of importance for the tool-in-use system They

are the plane of cut P se , the rake surface plane R s , and the clearance plane C s

For the purpose of determining the geometry of the active part of the cutting

tool, it is convenient to employ three reference planes P se , R s , and C s in

con-junction with the vector of the resultant cutting tool motion VΣ The current

orientation of the reference planes P se , R s , and C s is specified by unit normal

vectors nrs, ncs, and ce

Prior to running the analysis, it is necessary to represent equation rP =

rP=rP ( of the part surface P as well as equation r U P T =rT(U V T, T) of the

gen-erating surface T of the cutting tool in a common coordinate system X Y Z T T T

For this purpose, implementation of the operator Rs(T→P)of the resultant

coordinate system transformation is helpful Equations of tangent planes rP tp.

and rT tp. to the surfaces P and T at the point of interest M can be represented

The kinematic method can be employed for the derivation of the equation

of the surface of cut P se For this purpose, it is necessary to know the equation

of the cutting edge and the parameters of the resultant relative motion of the

cutting tool with respect to the work

The equation of the surface of cut P se can be obtained in the following

way Consider a form-cutting tool The cutting edge of the form-cutting tool

is determined as the line of intersection of the face rake surface R s by

clear-ance surface C s Therefore, in the coordinate system X Y Z T T T, the cutting

edge of the form-cutting tool can be described analytically by a set of two

260 Kinematic Geometry of Surface Machining

An auxiliary Cartesian coordinate system X ceY ceZce is associated with the

cutting edge Initially, axes of the coordinate system X ceYceZce align with

cor-responding axes of the coordinate system X TYTZT Then consider the motion

that the cutting edge together with the coordinate system X ceY ceZce is

per-forming in the coordinate system X TYTZT Parameters of this relative motion

of the cutting edge are identical to the corresponding parameters of motion

of the cutting tool relative to the work The equation of the cutting edge in a

current location of the coordinate system X ceY ceZce with respect to the

coordi-nate system X TYTZT can be represented in the form

Σ Σ

(6.69)

where ΞΣ designates the parameter of the resultant relative motion of the

cutting tool

On the premises of Equation (6.69), one of the two curvilinear parameters,

either the Urs or Vcs parameter can be expressed in terms of another parameter

For example, the Urs parameter is expressed in terms of the Vcs parameter This

relationship yields analytical representation in the form U rs=U V rs( cs)

Ulti-mately, this results in the vectorial equation of the surface of cut P se in the form

rse =rse[U V cs( cs),V cs,ΞΣ]=rse[V cs,ΞΣ] (6.70)

Similarly, the equation of the surface of cut P se can be expressed in terms of Vcs

and ΞΣ parameters For many purposes, the generating surface T of the

form-cutting tool can be considered as a good approximation to the surface of cut P se

In order to compose the tool-in-use system for machining a surface on a

conventional machine tool, two vectors are of principal importance: vector

VΣ of resultant relative motion of the cutting tool with respect to the work

and unit normal vector to the surface of cutting nse

The vector VΣ is computed from Equation (6.65) The unit normal nse

can be computed as the cross-product nse =use×vse For the derivation of

the unit tangent vectors use and vse , Equation (6.70) of the surface of cut P se

can be used That same unit normal vector nse can also be computed as the

cross-product (Figure 6.18):

where the unit vector vΣ is equal to vΣ = VΣ/|VΣ| Equation (6.71) for the

computa-tion of the unit normal vector nseis convenient for performing computations

Other equations for the computation of the unit normal vector nsecan be

The Geometry of the Active Part of a Cutting Tool 261

It is useful to keep in mind that the approximation nse≅nTis valid in most

practical cases of the computations Unit vectors vΣ (or VΣ) and nseare

helpful for the analytical representation of the tool-in-use system

6.3.3 reference Planes

Investigation of the impact of kinematics of a machining operation on actual

(kinematical) values of geometry of the active part of a cutting tool can be

traced back to research done by Pankin [6] or even to earlier works

A proper tool-in-use system is necessary but not sufficient for determining

geometric parameters of the active part of a cutting tool The specification of

the configuration of reference planes is also of critical importance

For free orthogonal cutting, the reference plane for the rake angle g, the

clearance angle a, the cutting wedge angle b, and the angle of cutting d is

the plane through the vector VΣ This reference plane is orthogonal to the

plane of cut P se For free oblique cutting, there are several reference planes

for specification of the angles g, a, b, and d.

The configuration of reference planes for nonfree cutting cannot be

speci-fied in general terms The mechanics of non-free cutting has not yet been

thoroughly investigated

6.3.3.1 The Plane of Cut Is Tangential to the Surface of Cut

at the Point of Interest M

For specification of the configuration of the plane of cut rse tp. , the vector of the

resultant motion VΣ of the cutting tool relative to the work, and the unit

vec-tor ce that is tangent to the cutting edge at M can be employed (Figure 6.19)

The Geometry of the Active Part of a Cutting Tool 263

cutting edge The angle of inclination λseis measured within the plane of cut

rse tp. This is the angle between the vector VΣ and the unit normal vector nce

The vector nce is orthogonal to the cutting edge (Figure 6.19a), and is within

the plane of cut rse tp. If observing from the end of the unit normal vector nse

to the surface of cut P se, then the positive angle λse is measured in a

counter-clockwise direction, and the negative angle λse is measured in a clockwise

direction (see Figure6.19b) When the equality λse= °0 is valid, then the

cut-ting is the orthogonal cutcut-ting Otherwise, when λse≠ °0 , then the more general

case of cutting — the oblique cutting — is observed Major frictions of the

cut-ting tool (that is, chip deformation, direction of chip flow over the rake surface,

etc.) depend upon the actual value of the angle of inclination λse

The algebraic value of the angle of inclination λsecan be computed from

the following equation (Figure6.19b):

For the cutting tools of various designs the optimal value of the angle of

inclination λsevaries within the interval λse= ± °80

6.3.3.2 The Normal Reference Plane

Configuration of the normal reference plane P ne of a cutting tool in the

tool-in-use system is identical to its configuration in the tool-in-hand system

The normal plane is orthogonal simultaneously to the rake surface R s, to

the clearance surface C s of the cutting wedge, to the plane of cut P se, and

ultimately, to the cutting edge (Figure 6.20) The unit normal vector nce to

the cutting edge is within the normal reference plane P ne Therefore,

configu-ration of the normal reference plane P ne can be specified in terms of any two

unit vectors nrs, ncs, nse, and nce at the point M (Figure6.20), or by the point

M and the unit vector ce along the cutting edge Evidently, there are many

more options for the specification of configuration of the normal reference

plane in the tool-in-use system rather than in the tool-in-hand system

6.3.3.2.1 Normal Rake Angle

Orientation of the rake surface of a cutting tool relative to the plane of cut

depends upon the actual value of normal rake angle γne The normal rake

angle is measured in the normal reference plane This is the angle that forms

the unit normal vector nse to the plane of cut P se and the rake surface R s The

value of the angle γne is measured from the vector nse toward the rake surface

Rs The normal rake angle γne is positive when the unit normal vector nsedoes

not pass through the cutting wedge of the tool, and it is negative when the

vec-tor nseis passing through the cutting wedge of the tool (Figure6.20b)

It is convenient to determine the normal rake angle γne as the angle that

complements to 90° the angle between the unit normal vectors nse and nrs

The Geometry of the Active Part of a Cutting Tool 265

6.3.3.2.2 Normal Clearance Angle

Orientation of the clearance surface C s with respect to the plane of cut P se

depends upon the normal clearance angle αne This angle is measured in the

normal reference plane The normal clearance angle αne is the angle that the

unit normal vector nse forms with the opposite direction of the unit normal

vector — the clearance surface C s The value of the clearance angle αne is

measured from the plane of cut P se toward the clearance surface C s The

normal rake angle αne is always positive (αne> °0) Only within a narrow

land along the cutting edge can the normal clearance angle αne be equal to

zero or even be negative (αne≤ °0 )

It is convenient to determine the normal clearance angle αne as the angle

that complements to 180° the angle between the unit vectors nse and ncs (see

For cutting tools of various designs, the optimal value of the normal

clear-ance angle αne is usually within the interval αne= ° ÷ °10 30

The uncut chip thickness a is the predominant factor that affects the

opti-mal value of the clearance angle On the premises of the analysis of impact of

chip thickness a, Larin [23] proposed an empirical formulae

αne

a

for the computation of reasonable value of the clearance angle

After a short period of cutting, a zero clearance angle αne= °0 is observed

within a narrow worn land along the cutting wedge

6.3.3.2.3 The Mandatory Relationship

For a workable cutting tool, satisfaction of the relationship N Nse⋅ ce <0 (or

the equivalent relationship n nse⋅ ce = -1 ) is necessary (see Figure6.20)

Vio-lation of the reVio-lationship is allowed only within a narrow land along the

cutting wedge

The normal cutting wedge angle is measured in the normal reference

plane The normal cutting wedge angle is the angle that forms the rake plane

Rs and the clearance plane C s The value of the angle βne can be computed

from a simple equation (see Figure6.20b):

βne= ° -90 (αne+γne) (6.78)The normal cutting angle is measured in the normal reference plane The

normal cutting angle is the angle that forms the plane of cut P se and the

clearance plane C s The value of this angle δne is equal (see Figure6.20b):

266 Kinematic Geometry of Surface Machining

Definitely, both the angles βne and δne can be expressed in terms of unit

normal vectors to the corresponding planes of the cutting wedge, and to the

reference surfaces

6.3.3.3 The Major Section Plane

Configuration of the major section plane P ve is determined by two directions

through the point M One of the directions is specified by the unit normal vector

nse to the plane of cut P se, and another direction is specified by the vector of the

resultant motion of the cutting tool VΣ with respect to the work (Figure 6.21a)

The major section plane P ve is perpendicular to the plane of cut P se

The equation of the major section plane P ve in terms of the vectors VΣ and

nseyields representation in vectorial form:

rve tp. -rse( )M [nse v ]

where rve tp. designates the position vector of a point of the major section plane

The rake angle γve is measured in the major section plane P ve (Figure6.21b)

The rake angle γve is equal to the angle between the unit normal vector nse to

the plane of cut, and the unit vector b is tangent to R s and is located within

the reference plane P ve:

The rake angle γve is positive when the vector nse does not penetrate the

cutting wedge, and it is negative when it does (Figure6.21b)

The clearance angle αve is the angle that the unit normal vector nse makes

with the unit vector c Here, the unit vector c is tangent to the line of intersection

of the clearance surface C s by the major section plane P ve(see Figure6.21b):

The angle of cutting δve is the angle that the unit vector b makes with the

vector VΣ of the resultant motion of the cutting tool relative to the surface of

268 Kinematic Geometry of Surface Machining

the cut (see Figure 6.21b):

When geometric parameters of the active part of a cutting tool are known

in the plane of cut P se and in the normal reference plane P ne, then the

corre-sponding geometric parameters can be computed in the major section plane

Pve, and vice versa Consider the computation of the rake angle γve as an

example of the proposed approach

The origin of the Cartesian coordinate system X Y Z T T T is at the point of

tangent to the line of intersection of the rake surface R s by the normal reference

λγλ

se ne se

1

(6.85)

The unit vector b is tangent to the line of intersection of the rake surface

R s by the major section plane P ve (see Figure6.21):

b=-

γγ

ν ν

=-

λ

interest M (see Figure 6.19) within the cutting edge Construct a vector A that is

(see Figure 6.19) It is equal

(see Figure 6.20) The projection length of vector A onto the coordinate

The Geometry of the Active Part of a Cutting Tool 269

By construction all three vectors A, b, ce are within a certain plane that

is tangent to the rake surface R s at the point of interest M This means that

the vectors A, b, ce represent a set of coplanar vectors Therefore, the triple

product of these vectors is identical to zero (A b c× ⋅ ≡e 0 Consequently, the )

following equality is valid:

0-

After the required formulae transformation are performed, one can come up

with the equations for the computation of the rake angle γve:

Following the way similar to that disclosed above, the equation

for the computation of the clearance angle αve can be derived

Equation (6.89) through Equation (6.91) for the computation of the rake

angle γveand the clearance angle αve are known since the publication by

Stabler [20]

The roundness r of the cutting edge in the major section plane Pve can be

computed from the equation ρve =ρne⋅cosλse Here ρne denotes the roundness

of the cutting edge in the normal reference plane P ne The equation for ρve is

another example of the correlation between the geometric parameters of the

active part of a cutting tool measured in different reference planes

6.3.3.5 The Main Reference Plane

P re is orthogonal to the vector VΣ of the resultant motion of the cutting tool

with respect to the surface of the cut (Figure 6.22) This reference plane can also

be determined as a plane through the unit normal vector nse to the surface of

cutting P se, and through the unit vector me that is orthogonal to the vector

VΣ The unit normal me belongs to the surface of cutting P se (Figure6.22) In

the coordinate system X Y Z T T T (see Figure6.22), the unit normal vector me is

identical to the unit vector i, i.e., of the X T-axis me=i

The Geometry of the Active Part of a Cutting Tool 271

where C( ) ϕe is a vector that aligns with the major cutting edge In the case

of a curved cutting edge, the vector C( ) ϕ is aligned with the tangent to the

cutting edge at the point of interest

6.3.3.5.2 The Minor Cutting Edge Approach Angle

Similarly, the minor cutting edge approach angle ϕ1e can be measured between

the projection of the minor cutting edge and the vector Vf of the feed-rate

motion (Figure6.23) The angle ϕ1e is also an acute angle (0° ≤ϕ1e≤ °90 )

Moreover, usually the value of the angle ϕ1e does not exceed the value of the

corresponding angle ϕe

For the computation of the minor cutting edge approach angle ϕ1e, the

fol-lowing formula can be employed:

where C( ϕ1e) denotes the vector that aligns with the minor cutting edge In

the case of a curved cutting edge, the vector C( ϕ1e) is aligned with the

tan-gent to the cutting edge at the point of interest

The angle ϕ1e is computed for a portion of the cutting edge within the

residual cusps On the rest of the portion of the cutting edge, it does not

affect the material removal process

In the event of small angles γne and λse, the analysis of actual values of

the angles ϕe and ϕ1e can be performed not in the main reference plane P se

but in the rake plane of the cutting tool In this case, instead of actual values

of the angles ϕe and ϕ1e, projections of these angles onto the rake plane can

Cutting edge angle k r and the minor (end) cutting edge angle k r 1 for a curved cutting edge in

the tool-in-use reference system.

272 Kinematic Geometry of Surface Machining

The tool tip (nose) angle εe is determined for the tip of a cutting tool The

angle εe can be computed from the equation

εe=180° -(ϕ ϕe+ 1e) (6.94)

The tip of a form-cutting tool coincides with the point of contact K of the

generating surface T of the cutting tool and the surface P being machined

(see Figure 6.23) At the point K the tool tip angle εe=180 °

The cutting edge approach angle ϕe affects the parameters of the uncut

chip, say of thickness a and width b of the uncut chip If the depth of cut t, and

the feed-rate V f (or S) are of constant value, then the following two equations

a S= ⋅sinϕe and b t= sinϕe are valid, and there will be a lower cutting edge

approach angle ϕe, a higher width b of the uncut chip, and a bigger tool-nose

angle εe

Both the angles ϕeand ϕ1eaffect parameters of residual cusps on the

machined part surface Bigger ϕeor ϕ1eresults in higher residual cusps on

the machined part surface

6.3.3.6 The Reference Plane of Chip Flow

In the case of free orthogonal cutting (when the inclination angle λse= °0 ),

the vector of chip motion over the rake surface is orthogonal to the cutting

edge Kinematical geometric parameters of the cutting edge are specified

in the plane that is orthogonal to the cutting edge The correctness of that

approach is comprehensively validated experimentally

Oblique cutting (when the angle of inclination λse≠ °0 ) is a much more

complex phenomenon than orthogonal cutting This is first because

deforma-tion of material does not occur in the major reference plane P m, but within

a certain volume, and thus deformation of material in a three-dimensional

space occurs Oblique cutting is much less understood than orthogonal

cutting

However, approximate results of the investigation of orthogonal cutting

can be adjusted for implementation for the analysis of oblique cutting as

well For oblique cutting, it is necessary to specify the rake angle taking into

consideration the direction of chip flow over the rake face

Lots of research has been carried out to determine the actual direction of

chip flow over the rake face The research was summarized by Stabler [20]

Without going into detail, consider Stabler’s chip flow law

6.3.3.6.1 Stabler’s Chip Flow Law

It is convenient to specify the direction of chip flow over the rake surface in

terms of the chip-flow angle h The chip-flow angle η is measured within the

rake plane This is the angle that the vector Vcf of the chip flow makes with

the perpendicular to the cutting edge within the rake plane (Stabler [20])

The Geometry of the Active Part of a Cutting Tool 273

The chip-flow angle η can be expressed in terms of width of cut b cf, width

b of the machined plane, and of inclination angle λse

cosη=b cosλ

b

cf

Equation (6.95) is derived under the assumption that there is no deformation

within the chip width It is proven that this assumption is valid for

orthogo-nal cutting [21]

In compliance with the chip-flow law, the chip-flow angle η is approximately

equal to the angle of inclination of the cutting edge λse This correlation can be

analytically expressed by the following approximate equality η λ≅ se The

equa-tion is based on the assumpequa-tion that b cf =b, and it is valid for all cutting tools

having the inclination angle λse< °45 When the inclination angle λse≥ °45 , then

the difference between the angles λseand h remains within (λse- ≤ ÷ °η) 5 6

Stabler later modified the chip-flow law and represented it in the form

η≅( ,1 0 0 9÷ , )λse

Along with the chip-flow angle h, the direction of chip flow over the rake

face can be specified by the projection of this angle onto the main reference

plane [20]

6.3.3.6.2 The Chip-Flow Rake Angle

In an attempt to specify (approximately) the poorly understood oblique

cut-ting in terms of the comprehensively investigated orthogonal cutcut-ting, the

term chip-flow reference plane was introduced The chip-flow rake angle γcf is

measured in the chip-flow reference plane P cf The rake angle γcf (as well

as the depth of cut t cf) in the chip-flow reference plane differs from the

ana-logue parameters measured in other reference planes

The chip-flow reference plane P cf is the plane through the vectors VΣ and

Vcf Here VΣ designates the vector of resultant motion of the cutting edge

with respect to the surface of the cut, and Vcf designates the vector of chip

flow over the rake surface The vector Vcf is located within the rake plane It

forms the chip-flow angle h perpendicular to the cutting edge (Figure 6.24)

The vector Vcf is orthogonal to the unit normal vector nrs to the rake

sur-face R s Therefore, the equality V ncf ⋅ rs=0 occurs

At the point of interest, the chip-flow reference plane P cf is a plane through

the vectors VΣ and Vcf at the point M This yields

(rcf -r( )M)⋅VΣ×Vcf =0 (6.96)

Here rcf designates the position vector of a point of the chip-flow reference

plane P cf, and r( )M designates the position vector of the point of interest M.

The chip-flow rake angle γcf is the angle that the vector Vcf of chip flow

over the rake plane forms with the main reference plane P re (Figure 6.20)